Let $A = \{1, 2, 3, 4, 5, 6, 7\}$. Then the relation $R = \{(x, y) \in A \times A : x + y = 7\}$ is

The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c), (b, d)\}$ on the set $\{a, b, c, d\}$ so that it is an equivalence relation,is $.........$

Show that the relation $R$ defined in the set $A$ of all triangles as $R = \{(T_{1}, T_{2}) : T_{1} \text{ is similar to } T_{2}\}$,is an equivalence relation. Consider three right-angled triangles $T_{1}$ with sides $3, 4, 5$,$T_{2}$ with sides $5, 12, 13$,and $T_{3}$ with sides $6, 8, 10$. Which triangles among $T_{1}, T_{2}$,and $T_{3}$ are related?

Let $A = \{2, 3, 4, 5, \ldots, 16, 17, 18\}$. Let $R$ be the relation on the set $A \times A$ defined by $(a, b) R (c, d)$ if and only if $ad = bc$ for all $(a, b), (c, d) \in A \times A$. Then,the number of ordered pairs in the equivalence class of $(3, 2)$ is:

If $A = \{x \in Z^+ : x < 10\}$ and $x$ is a multiple of $3$ or $4$,where $Z^+$ is the set of positive integers,then the total number of symmetric relations on $A$ is

Let $A = \{1, 2, 3, 4\}$ and $R = \{(1, 2), (2, 3), (1, 4)\}$ be a relation on $A$. Let $S$ be the smallest equivalence relation on $A$ such that $R \subset S$. If the number of elements in $S$ is $n$,then the value of $n$ is: